Cylindrical algebraic decomposition I: the basic algorithm

Given a set of r-variate integral polynomials, a cylindrical algebraic decomposition (cad) of euclidean r-space E^r partitions E^r into connected subsets compatible with the zeros of the polynomials. Each subset is a cell. G. E. Collins gave a cad construction algorithm in 1975, as part of a quantifier elimination procedure for real closed fields. The cad algorithm has found diverse applications (optimization, curve display); new applications have been proposed (term rewriting systems, motion planning). In the present two-part paper, the authors give an algorithm which determines the pairs of adjacent cells as it constructs a cad of E². Such information is often useful in applications. In Part I they describe the essential features of the r-space cad algorithm, to provide a framework for the adjacency algorithm in Part II.